By Nyulaszi L.
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The only difference is that the time dependent waves become time independent standing waves. If one solves for the Green’s function in coordinate representation, one gets : √ 1 ei Ei |r−r | G0 (r − r ; Ei ) = − . 2 Hamiltonians and formal multiple scattering theory 27 where V (r) is the (non-point) scatterer. Eq. 11) is the well known LippmanSchwinger equation. Now we leave this part and try to make a little bit more formal treatment. 2 Hamiltonians and formal multiple scattering theory We are going to formulate this in terms of Hamiltonians, which turn up naturally in discussions about Green’s functions.
The reason for fixing the kinetic energy to zero is that it was found that both the structure constants and phase shifts were strongly energy dependent, and it was further found out  that the wave function depends on κ through the wavelength 2π/κ, which is a much larger wavelength than the length scales of the potential. The strong κ-dependence was therefore thought to be unphysical, and also disappeared at the solutions of the KKR equations . It was then decided to fix κ to E − v0 and thereby get rid of the interstitial.
This is the first of the original free-electron energy values, which has been shifted towards higher energies by the screening employed for the screened spherical waves. The large absolute value of the slope matrix here introduces numerical instabilities in the inversion routines used for the screening, and makes calculations impossible. Our idea was to calculate the slope matrix a bit out in the complex plane instead, where the absolute value is much smaller. This indeed brings an improvement in that it is possible to actually calculate the slope matrix, and in that the expansion generally is slightly better if the expansion center is placed in the center of the energy contour.